Find the coefficient of `x^n` in the polynomial `(x+^n C_0)(x+3^n C_1)xx(x+5^n C_2)[x+(2n+1)^n C_n]dot`
1 Answers
Correct Answer - `(n+1)2^(n)`
Given polynomial is `n +1` degree polynomial.
`(x+.^(n)C_(0))(x+3.^(n)C_(2))(x+5.^(n)C_(2))"....."[x+(2n+1).^(n)C_(n)]`
`= x^(n+1)+a_(1)x^(n)+"...."+a_(n+1)`
Then,
`- (a_(1))/(a_(0))=-.^(n)C_(0)-3.^(n)C_(1)-5.^(n)C_(2)-"....."-(2n+1).^(n)C_(n)`
`rArr a_(1)=.^(n)C_(0)+3.^(n)C_(1)+5.^(n)C_(2)-"......"+(2n+1).^(n)C_(n)`
`= underset(r=0)overset(n)sum.^(n)C_(r)(2r+1)=2underset(r=0)overset(n)sumr^(n)C_(r)+underset(r=0)overset(n)sum.^(n)C_(r)`
`= 2underset(r=0)overset(n)sumn..^(n-1)C_(r-1) + underset(r=0)overset(n)sum.^(n)C_(r)`
`= 2n 2^(n+1) + 2^(n) = (n+1)2^(n)`
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